The study characterizes channel surfaces in Lie sphere geometry and their transformations.
problem Characterizing channel surfaces in Lie sphere geometry.
method Using Ω0-surfaces and Dupin cyclide congruences to study transformations and Ribaucour pairs. result Characterization of channel surfaces and their transformations.
Study integrable discretizations of cyclic systems with circular coordinate lines.
problem Integrable discretizations of 3D cyclic systems with circular coordinate lines.
method Investigate circle congruences and flat connections in the context of discrete cyclic systems.
result Characterization of circle congruences and existence of certain flat connections.
Geometrically revisits Dupin cyclidic systems using evolving circles and cyclides.
problem Understanding the geometric properties and evolution of Dupin cyclidic systems.
method Evolving initial circles or Dupin cyclides to generate Lamé families of Dupin cyclidic systems in various space forms.
result Lamé families are parallel surfaces in different space forms.
Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth C1-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit de…
This article is devoted to the study of cyclides osculating general surfaces. We show that generically, at any point of a surface, one has a one-parameter family of cyclides tangent to a surface curve of order three and among them just one is tangent to this curve of order four. This one will be called the osculating c…
The method of moving frames in Lie sphere geometry has produced significant results in the classification of Dupin hypersurfaces in spheres. What is the secret of its effectiveness? The answer emerges in the classification of nonumbilic isoparametric surfaces in the space form geometries. Using the method of moving fra…
Supercyclides are surfaces with a characteristic conjugate parametrization consisting of two families of conics. Patches of supercyclides can be adapted to a Q-net (a discrete quadrilateral net with planar faces) such that neighboring surface patches share tangent planes along common boundary curves. We call the result…
Of all real Lagrangian--Grassmannians LG(n,2n), only LG(2,4) admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space S1,2. Using Cartan's method of moving frames, we study hyperbolic (timelike) surfaces in LG(2,4) modulo the conformal symplectic gro…
Square Clifford torus uniquely determined by isoperimetric ratio, rectangular torus not.
problem Uniqueness of 3D shape of rectangular Clifford torus based on isoperimetric ratio.
method Closed-form formulas for isoperimetric ratio of stereographic projection, strict monotonicity.
result Isoperimetric ratio does not uniquely determine rectangular Clifford torus shape.
Lie sphere geometry helps classify Dupin hypersurfaces.
problem Classifying Dupin hypersurfaces in Lie sphere geometry.
method Generalizing Dupin hypersurfaces to Lie sphere geometry and using Lie sphere transformations.
result Many classifications of proper Dupin hypersurfaces have been obtained.
Similar to the definition of Dupin hypersurface in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using the framework of conformal geometry. Further we classify the spacelike Dupin hype…
We introduce the notion of weak reduciblity for Dupin submanifolds with arbitrary codimension. We give a complete characterization of all weakly reducible Dupin submanifolds, as a consequence of a general result on a broader class of Euclidean submanifolds. As a main application, we derive an explicit recursive procedu…
Survey of Dupin hypersurfaces in Lie sphere geometry.
problem Classifying Dupin hypersurfaces in Lie sphere geometry.
method Detailed description of Lie sphere geometry concepts and classification results.
result Many classification results relating Dupin hypersurfaces to isoparametric hypersurfaces in spheres.
Survey on Dupin hypersurfaces and isoparametric hypersurfaces in spheres.
problem Understanding the relationship between compact proper Dupin hypersurfaces and isoparametric hypersurfaces in spheres.
method Survey of existing results and related developments.
result Progress on Cecil and Ryan's conjecture on compact proper Dupin hypersurfaces.
In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat L…
We discuss the problem of R-separability (separability of variables with a factor R) in the stationary Schrödinger equation on n-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of R-separability in PDE (Laplace equation on E3…
Compact Dupin hypersurfaces without constant Lie curvatures found.
problem Finding compact Dupin hypersurfaces with non-constant Lie curvatures.
method Two constructions of compact proper Dupin hypersurfaces in Sn. result Examples of compact proper Dupin hypersurfaces without constant Lie curvatures.
New findings on Chern's conjecture for Dupin hypersurfaces.
problem Chern's conjecture for hypersurfaces with constant scalar curvature.
method Combining topology and geometry to reduce assumptions in algebraic arguments.
result Closed proper Dupin hypersurfaces with specific conditions are isoparametric.
The paper updates methods for studying proper Dupin hypersurfaces in Lie sphere geometry.
problem Understanding proper Dupin hypersurfaces in Lie sphere geometry.
method Using moving frames to study proper Dupin hypersurfaces locally.
result Obtained several classification theorems of proper Dupin hypersurfaces.
New examples of austere submanifolds and hypersurfaces with specific curvature properties.
problem Local classification of proper Dupin hypersurfaces.
method Construction of austere submanifolds in unit spheres.
result Found three irreducible proper Dupin hypersurfaces with 5 distinct principal curvatures.
We introduce two basic invariant forms which define generic surface in 3-space uniquely up to Lie sphere equivalence. Two particularly interesting classes of surfaces associated with these invariants are considered, namely, the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For diagonally-cyclidic surfaces …
We show that every spherical 2-Dupin submanifold that is not a hypersurface is conformally congruent to the standard embedding of the real, complex, quaternionic or octonionic projective plane. We also classify 2-CPC, 2-umbilical and weakly 2-umbilical submanifolds in space forms.
Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.
problem Discretizing surfaces with spherical curvature lines.
method Lie-geometric discretisation in terms of principal contact element nets.
result Circular nets with two families of spherical parameter lines are related to Darboux cyclides.
The paper studies special surfaces in pseudo-Euclidean space.
problem Characterizing and classifying ε-isothermic surfaces in pseudo-Euclidean 3-space. method Analyzing the pseudo-Calapso equation and providing explicit coordinates for Dupin surfaces.
result Explicit solutions to the pseudo-Calapso equation are provided.
We prove that any connected proper Dupin hypersurface in Rn is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in Rn that satisfies a certain finiteness condition. Hence any taut submanifo…
Study on braid group quotients by congruence subgroups.
problem Understanding the image of congruence subgroups in GL(n,Z).
method Characterization through symplectic congruence subgroups.
result Open problem solved: image of congruence subgroups in GL(n,Z).
A multi-neck spacetime wormhole is constructed with a simple metric tensor.
problem Existence of multi-neck spacetime wormholes.
method Spherical inversion of a 3-torus to create a 3-neck spacetime wormhole.
result Exact solution of Einstein's field equations for a multi-neck spacetime wormhole.
If M is an isoparametric hypersurface in a sphere Sn with four distrinct principal curvatures, then the principal curvatures κ1,...,κ4 can be ordered so that their multiplicities satisfy m1=m2 and m3=m4, and the cross-ratio r of the principal curvatures (the Lie curvature) equals -1. In this paper, w…
Enumerated all principal congruence link complements in 3D space.
problem Identifying all principal congruence link complements in S3. method Enumerated all principal congruence link complements in S3. result Answered a question posed by W. Thurston about principal congruence link complements.
In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under {appropriate} conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of t…
The paper describes the geometric properties of line congruences' singularities.
problem Understanding the singularities of generic line congruences.
method Use of an equiaffine pair to define generic line congruences.
result Geometric description of folds, cusps, and swallowtails as singularities of generic line congruences.
The paper classifies singularities of line congruences in 4D space.
problem Classifying singularities of line congruences in 4D space.
method Generic classification approach for 3-parameter line congruences and Blaschke normal congruences.
result Generic classification of singularities of 3-parameter line congruences in R4. The works of William Rowan Hamilton in Geometrical Optics are presented, with emphasis on the Malus-Dupin theorem. According to that theorem, a family of light rays depending on two parameters can be focused to a single point by an optical instrument made of reflecting or refracting surfaces if and only if, before ente…
Study on braid groups' congruence subgroups and their crystallographic quotients.
problem Understanding congruence subgroups and crystallographic quotients of braid groups.
method Investigation of lower central series of congruence braid groups related to B3. result Quotients of congruence braid groups are almost crystallographic.
The paper classifies singularities of plane congruences and affine distance functions.
problem Classifying singularities of plane congruences and affine distance functions.
method Classification through 2-parameter plane congruences in \(\mathbb{R^4}\) and affine normal plane congruences.
result Generic singularities of plane congruences and affine distance functions are classified.
Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3-manifolds generated by Dehn surgery with denominator f: weak f-congruence, f-congruence, and strong f-congruence. If f is odd, weak f-congruence preserves the ring structure on cohomology with Z_f-coefficients. We…
This paper explores geometric insights into discrete R-congruences and their envelopes.
problem Understanding the ambiguity in discrete R-congruences and their envelopes.
method Analyzes discrete R-congruences that are enveloped by specific types of surfaces and maps.
result Discovers a 2-parameter family of discrete enveloping surfaces for discrete R-congruences.
We prove some new rigidity results for proper biharmonic immersions in Sn of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces satisfying intrinsic properties; PMC submanifolds; parallel submanifolds.
Paper explores relations between braid groups and their quotients.
problem Understanding relations between braid groups and their quotients.
method Recalling and introducing elements of congruence braid groups, establishing isomorphisms between crystallographic and congruence braid groups.
result Established isomorphisms between crystallographic braid groups and quotients of congruence braid groups.
New BDEs reveal singular surfaces from line congruences.
problem Understanding binary differential equations associated with line congruences.
method Applied pointwise to quadratic differential forms, studying quotients of quadratic forms and associated polar lines.
result Introduced a new singular surface in Euclidean 3-space.
Technical report classifies all principal congruence link groups.
problem Classifying all principal congruence link complements in S^3.
method Comprehensive case analysis, necessary computations, and computer programs.
result Complete classification of all principal congruence link groups.
The study explores congruence subgroups of braid groups and their quotients.
problem Understanding the structure of congruence subgroups of braid groups.
method Utilizing the integral Burau representation and results from integral matrices, the study examines quotients of these subgroups.
result Findings of quotients that are not isomorphic to symmetric groups.
Criterion for congruence RFRS towers in hyperbolic lattices.
problem Criteria for hyperbolic lattices to have congruence RFRS towers.
method Criterion based on congruence subgroups.
result Virtually fibered Bianchi groups and new RFRS Kähler groups.
Study of line congruences for Appell's rank-4 hypergeometric functions.
problem Understanding line congruences for Appell's rank-4 hypergeometric functions.
method Derived original formulae for Laplace transform of rank-4 system, applied to geometry of surfaces defined by these functions.
result Natural line congruences for Laplace transforms of Appell's rank-4 functions form a W-congruence.
This paper explores dual relations between line congruences and surfaces in 3D and 4D.
problem Understanding the duality between line congruences in R3 and surfaces in R4. method Analyzes the correspondence between principal lines and asymptotic lines, ridge curves and flat ridge curves, and subparabolic curves.
result Discusses the behavior of subparabolic curves at the discriminant curve of the line congruence and the parabolic curve of the dual surface.
Abstract lists known link diagrams for all principal congruence link complements.
problem Identifying all known link diagrams for principal congruence link complements.
method Compilation of known link diagrams.
result Compilation of known link diagrams for principal congruence link complements.
DCL method improves congruency in machine learning tasks.
problem Varying task-specific semantics in training data.
method Direction Concentration Learning (DCL) method to enhance congruency.
result Improves performance in saliency prediction, continual learning, and classification tasks.
Characterizes W-congruences to study their stable umbilical points.
problem Understanding singularities of W-congruences.
method Characterization of W-congruences to study umbilical points.
result Examples of isolated stable umbilical points of type Am.